Weierstrass functions: Information from Answers.com

In mathematics, the Weierstrass functions are special functions of a complex variable that are auxiliary to the Weierstrass elliptic function. They are named for Karl Weierstrass.

1 Weierstrass sigma-function
2 Weierstrass zeta-function
3 Weierstrass eta-function
4 Weierstrass p-function
5 See also

Weierstrass sigma-function

The Weierstrass sigma-function associated to a two-dimensional lattice $\Lambda\subset\Complex$ is defined to be the product

$\sigma(z;\Lambda)=z\prod_{w\in\Lambda^{*}} \left(1-\frac{z}{w}\right) e^{z/w+\frac{1}{2}(z/w)^2}$

where $Λ *$ denotes $Λ - {0}$ .

Weierstrass zeta-function

The Weierstrass zeta-function is defined by the sum

$\zeta(z;\Lambda)=\frac{\sigma'(z;\Lambda)}{\sigma(z;\Lambda)}=\frac{1}{z}+\sum_{w\in\Lambda^{*}}\left( \frac{1}{z-w}+\frac{1}{w}+\frac{z}{w^2}\right).$

Note that the Weierstrass zeta-function is basically the logarithmic derivative of the sigma-function. The zeta-function can be rewritten as:

$\zeta(z;\Lambda)=\frac{1}{z}-\sum_{k=1}^{\infty}\mathcal{G}_{2k+2}(\Lambda)z^{2k+1}$

where $\mathcal{G}_{2k+2}$ is the Eisenstein series of weight $2 k + 2$ .

Also note that the derivative of the zeta-function is $-\wp(z)$ , where $\wp(z)$ is the Weierstrass elliptic function

The Weierstrass zeta-function should not be confused with the Riemann zeta-function in number theory.

Weierstrass eta-function

The Weierstrass eta-function is defined to be

$\eta(w;\Lambda)=\zeta(z+w;\Lambda)-\zeta(z;\Lambda), \mbox{ for any } z \in \Complex$

It can be proved that this is well-defined, i.e. $ζ(z + w;Λ) - ζ(z;Λ)$ only depends on w. The Weierstrass eta-function should not be confused with the Dedekind eta-function.

Weierstrass p-function

The Weierstrass p-function is defined to be

$\wp(z;\Lambda)= -\zeta'(z;\Lambda), \mbox{ for any } z \in \Complex$

The Weierstrass p-function is an even elliptic function of order N=2 with a double pole at each lattice and no others.

	Equianharmonic
	Quasiperiodic function
	Lemniscatic elliptic function

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Weierstrass functions

Contents

Weierstrass sigma-function

Weierstrass zeta-function

Weierstrass eta-function

Weierstrass p-function

See also